2024/04/21 by Sampaio, José Edson, da Silva, Euripedes Carvalho
#53C12 #53C42 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2404.13772
The main goal of this present paper is to bring the results proved by Barbosa, Kenmotsu and Oshikiri (1991) and its ideas to a perspective where the Ricci curvature is bounded from below. For instance, for a foliation by CMC hypersurfaces on a compact (without boundary) Riemannian manifold Mn+1 with Ricci curvature bounded from below by -nK0≤ 0 and such that the mean curvature H of the leaves of the foliation satisfies |H|≥ √(K0), we prove that |H|≡ √(K0) and all the leaves are totally umbilical. This gives, in particular, a generalization for the result proved by Barbosa, Kenmotsu and Oshikiri (1991), where the above result was proved in the case K0=0. We also obtain a proof of the following: for a foliation by CMC hypersurfaces on a compact (without boundary) Riemannian manifold M with Ricci curvature bounded from below by -nK0≤ 0, the mean curvature H of the leaves of the foliation satisfies |H|≤ √(K0). Furthermore, if the foliation contains a leaf L whose absolute mean curvature is |HL|=√(K0), then either K0=0 and all the leaves of \mathfrakF are totally geodesic, or K0>0 and there is a totally umbilical leaf.